Transport of structure in higher homological algebra

The Karoubi envelope and weak idempotent completion of an extriangulated category

“…For an object A of C, we have that i C (A) = (A, 1 A ) and for a morphism f in C, we have that i C ( f ) = f . We will show that this functor is an extriangulated functor in the sense of [2,Definition 2.31]. In particular, the functor i C preserves the extriangulated structure of C. For δ ∈ E(C, A), we have that E( f , g)(δ) = f * g * δ.…”

Recollements of extriangulated categories

“…The extriangulated functor between two extriangulated categories has been defined in [3]. For our requirements, we modify the definition as follows: Definition 2.13.…”

s-Recollements and its localization

“…Under the assumption of the (WIC) condition, Wang-Wei-Zhang [35] introduced the notion of compatible morphisms. Using this notion, they modified the extriangulated functor between two extriangulated categories, which has been defined in [5], and then, introduced the concepts of right (left) exact functors in extriangulated categories. By these frameworks, Wang-Wei-Zhang [35] gave a simultaneous generalization of recollements of abelian categories and triangulated categories, which we call the recollement of extriangulated categories.…”

“…Without the (WIC) condition, we introduce a new concept, called s-recollements of extriangulated categories. Moreover, in this definition, we do not need to modify the extriangulated functor (in this paper, we call exact functor) in the sense of [5]. Because of this point, the notion of s-recollements looks more natural.…”